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Beliebt Rechnen >

integral of (2x^2-2x-8)/(x^2(x-4))

Lösung

\int \frac{2 x ^{2} - 2 x - 8}{x ^{2} ( x - 4 )} d x

Lösung

ln ∣ x ∣ + ln ∣ x - 4 ∣ - \frac{2}{x} + C

Schritte zur Lösung

\int \frac{2 x ^{2} - 2 x - 8}{x ^{2} ( x - 4 )} d x

Multipliziere aus

\frac{2 x ^{2} - 2 x - 8}{x ^{2} ( x - 4 )} : \frac{2 x ^{2}}{x ^{2} ( x - 4 )} - \frac{2 x}{x ^{2} ( x - 4 )} - \frac{8}{x ^{2} ( x - 4 )}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2 x ^{2}}{x ^{2} ( x - 4 )} d x - \int \frac{2 x}{x ^{2} ( x - 4 )} d x - \int \frac{8}{x ^{2} ( x - 4 )} d x

\int \frac{2 x ^{2}}{x ^{2} ( x - 4 )} d x = 2 ln ∣ x - 4 ∣

\int \frac{2 x}{x ^{2} ( x - 4 )} d x = 2 (- \frac{1}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x - 4 ∣)

\int \frac{8}{x ^{2} ( x - 4 )} d x = 8 (- \frac{1}{16} ln ∣ x ∣ + \frac{1}{4 x} + \frac{1}{16} ln ∣ x - 4 ∣)

= 2 ln ∣ x - 4 ∣ - 2 (- \frac{1}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x - 4 ∣) - 8 (- \frac{1}{16} ln ∣ x ∣ + \frac{1}{4 x} + \frac{1}{16} ln ∣ x - 4 ∣)

Vereinfache

2 ln ∣ x - 4 ∣ - 2 (- \frac{1}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x - 4 ∣) - 8 (- \frac{1}{16} ln ∣ x ∣ + \frac{1}{4 x} + \frac{1}{16} ln ∣ x - 4 ∣) : ln ∣ x ∣ + ln ∣ x - 4 ∣ - \frac{2}{x}

= ln ∣ x ∣ + ln ∣ x - 4 ∣ - \frac{2}{x}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + ln ∣ x - 4 ∣ - \frac{2}{x} + C

Graph

Beliebte Beispiele

limit as x approaches 0-of cos(1/x)

x \to 0 - lim (cos (\frac{1}{x}))

integral of 1/(1^2+1x^2)

\int \frac{1}{1 ^{2} + 1 x ^{2}} d x

derivative of (x^2+100/x)

\frac{d}{d x} (\frac{x ^{2} + 100}{x})

limit as x approaches 2 of ln(2)

x \to 2 lim (ln (2))

derivative of-1/(x^4-5/(x^2))

\frac{d}{d x} (- \frac{1}{x ^{4}} - \frac{5}{x ^{2}})